Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. It is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. (These special situations are eliminated in a companion paper by the first author.) An example coming from the exceptional group E6(q) shows that the kernel of the action of Gx on the ball of radius 2 around x can be non-trivial.

It was proved by Weyl [8] in 1916 that the sequence of values of αn2 is uniformly distributed modulo 1, for any fixed real irrational α. Indeed this result covered sequences αnd for any fixed positive integer exponent d. However Weyl's work leaves open a number of questions concerning the finer distribution of these sequences. It has been conjectured by Rudnick, Sarnak and Zaharescu [6] that the fractional parts of αn2 will have a Poisson distribution provided firstly that α is “Diophantine”, and secondly that if a/q is any convergent to α then the square-free part of q is q1+o(1). Here one says that α is Diophantine if one has(1.1)for every rational number a/q and any fixed ϵ > 0. In particular every real irrational algebraic number is Diophantine. One would predict that there are Diophantine numbers α for which the sequence of convergents pn/qn contains infinitely many squares amongst the qn. If true, this would show that the second condition is independent of the first. Indeed one would expect to find such α with bounded partial quotients.

For P ⊂ 3 a finite volume geodesic polyhedron, with the property that all interior angles between incident faces are of the form π/mij (mij ≥ 2 an integer), there is a naturally associated Coxeter group ΓP. Furthermore, this Coxeter group is a lattice inside the semi-simple Lie group O+(3, 1) = Isom(3), with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such groups ΓP in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate the K−1 and Wh of the groups Dn and Dn × 3, and we also summarize what is known about the 0.

We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the standard argument for the nonsingular case, involving Riemann–Roch without denominators. The new approach relies on the explicit computation of an ideal, and a mild generalization of a well-known formula for the normal bundle of a proper transform ([8, B·6·10]).

We also discuss alternative, very short proofs of the standard formula in some cases: an approach relying on the theory of Chern–Schwartz–MacPherson classes (working in characteristic 0), and an argument reducing the formula to a straightforward computation of Chern classes for sheaves of differential 1-forms with logarithmic poles (when the center of the blow-up is a complete intersection).

Let X be an irreducible smooth complex projective curve of genus g ≥ 2, and let x ∈ X be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, φ), where E is coherent sheaf on X of rank r and fixed determinant ξ, and φ: Ex → r is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ–semistable framed bundles τ. We prove a Torelli theorem for τ, for τ > 0 small enough, meaning, the isomorphism class of the one–pointed curve (X, x), and also the integer r, are uniquely determined by the isomorphism class of the variety τ.

Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.

There are many constructions of the Suzuki groups in the literature (see for example Suzuki's original paper [3], as well as [1, 2, 4]), and one needs to make a strong case to justify publishing another. Yet I believe the construction below is sufficiently new and sufficiently elementary that in time it will come to be regarded as the standard construction.

For C*-algebras A1, A2 the map (I1, I2) → ker(qI1 ⊗ qI2) from Id′(A1) × Id′(A2) into Id′(A1 ⊗minA2) is a homeomorphism onto its image which is dense in the range. Here, for a C*-algebra A, the space of all proper closed two sided ideals endowed with an adequate topology is denoted Id′(A) and qI is the quotient map of A onto A/I. This result is used to show that any continuous function on Prim(A1) × Prim(A2) with values into a T1 topological space can be extended to Prim(A1 ⊗minA2). This enlarges the scope of [7, corollary 3·5] that dealt only with scalar valued functions. A new proof for a result of Archbold [3] about the space of minimal primal ideals of A1 ⊗minA2 is obtained also by using the homeomorphism mentioned above. New proofs of the equivalence of the property (F) of Tomiyama for A1 ⊗minA2 with certain other properties are presented.

Let f(x) ∈ [x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1/r1, r2, 1/r2, . . ., rd, 1/rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 + 1/r1, . . ., rd + 1/rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg ≅ Gf/N, where N is isomorphic to a subgroup of C2d. In a naive sense, the generic case is Gf ≅ C2d ⋊ Sd, with N ≅ C2d and Gg ≅ Sd. When f(x) has extra structure this may be reflected in the Galois group, and it is not always true even that Gf ≅ N ⋊ Gg. For example, for cyclotomic polynomials f(x) = Φn(x) it is known that Gf ≅ N ⋊ Gg if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4.

In this paper we deal with irreducible reciprocal monic polynomials f(x) ∈ [x] that are ‘close’ to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f(x) has degree at least 4, this means that f(x) is the minimal polynomial of a Salem number. We show that in this case one always has Gf ≅ N ⋊ Gg, and moreover that N ≅ C2d or C2d−1, with the latter only possible if d is odd.

Let L be a finitely generated Lie algebra which is a split extension of a free nilpotent Lie algebra N by a finite dimensional abelian Lie algebra. Let V denote the quotient of N by its commutator subalgebra; we can regard V as a module for L/N. We discuss the relationship between the homological finiteness properties of V and those of L. In particular, we show that if L has type FPm and N has class c then V is 1 + c(m − 1)-tame (equivalently, the (1 + c(m − 1))th tensor power of V is finitely generated under the diagonal action of L/N).

Let G be a compact, connected, abelian group with dual group Γ. The set E ⊂ has zero discrete harmonic density (z.d.h.d.) if for every open U ⊂ G and μ ∈ Md(G) there exists ν ∈ Md(U) with = on E. I0 sets in the duals of these groups have z.d.h.d. We give properties of such sets, exhibit non-Sidon sets having z.d.h.d., and prove union theorems. In particular, we prove that unions of I0 sets have z.d.h.d. and provide a new approach to two long-standing problems involving Sidon sets.

We study generalized symbolic powers and form ideals of powers and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.

We give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closures of (cabled) string links. We show that finite type invariants classify string links up to Ck-moves for k ≤ 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar classification of string links up to Ck-moves and concordance for k ≤ 6.

We study the squaring operation Sq0 on the dual of the minimal -generators of the Dickson algebra. We show that this squaring operation is isomorphic on its image. We also give vanishing results for this operation in some cases. As a consequence, we prove that the Lannes–Zarati homomorphism vanishes (1) on every element in any finite Sq0-family in except possibly the family initial element, and (2) on almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes.

Let p be a prime number and M a quadratic number field, M ≠ ℚ() if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/ℚ with Galois group D2p and an elliptic curve E/ℚ such that F contains M and the p-Selmer group of E/F has size at least pd.

We study the arc and curve complex AC(S) of an oriented connected surface S of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of AC(S) coincides with the natural image of the extended mapping class group of S in that group. We also show that for any vertex of AC(S), the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in S that represents that vertex. We also give a proof of the fact if S is not a sphere with at most three punctures, then the natural embedding of the curve complex of S in AC(S) is a quasi-isometry. The last result, at least under some slightly more restrictive conditions on S, was already known. As a corollary, AC(S) is Gromov-hyperbolic.

We give an elementary construction of the compact real form of the Lie algebra 2. This construction exhibits the group 23·L3(2) as a group of automorphisms. We also show that there is a unique 14-dimensional real Lie algebra invariant under the action of this group.

We show that when r ≥ 5 is prime, the SO(3) Witten–Reshetikhin–Turaev quantum invariants for three-manifolds at the level r form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.

The radial index of a 1-form on a singular set is a generalization of the classical Poincaré–Hopf index. We consider different classes of closed singular semi-analytic sets in n that contain 0 in their singular locus and we relate the radial index of a 1-form at 0 on these sets to Poincaré–Hopf indices at 0 of vector fields defined on n.

We derive some necessary conditions on a Riemannian metric (M, g) in four dimensions for it to be locally conformal to Kähler. If the conformal curvature is non anti–self–dual, the self–dual Weyl spinor must be of algebraic type D and satisfy a simple first order conformally invariant condition which is necessary and sufficient for the existence of a Kähler metric in the conformal class. In the anti–self–dual case we establish a one to one correspondence between Kähler metrics in the conformal class and non–zero parallel sections of a certain connection on a natural rank ten vector bundle over M. We use this characterisation to provide examples of ASD metrics which are not conformal to Kähler.

We establish a link between the ‘conformal to Kähler condition’ in dimension four and the metrisability of projective structures in dimension two. A projective structure on a surface U is metrisable if and only if the induced (2, 2) conformal structure on M = TU admits a Kähler metric or a para–Kähler metric.